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On families of periodic solutions of the restricted three-body problem

  • Alexander D. Bruno
  • Victor P. Varin
Conference paper

Abstract

We consider the plane circular restricted three-body problem. It is described by an autonomous Hamiltonian system with two degrees of freedom and one parameter μ ε [0, 1/2] which is the mass ratio of the two massive bodies. Periodic solutions of this problem form two-parameter families. We propose methods of computation of symmetric periodic solutions for all values of the parameter μ. Each solution has a period and two traces, namely, the plane and the vertical one. Two characteristics of a family, i.e., its intersection with the symmetry plane, are plotted in the four coordinate systems used in the investigations: two global and two local ones related to the two massive bodies. We also describe generating families, i.e., the limits of families as μ → 0, which are known almost explicitly. As an example, we consider the family h, which begins with retrograde circular orbits of infinitely small radius around the primary P 1 of bigger mass. For this family, we cite detailed data for μ = 0 and μ ≈ 10−3 and give a brief description of its evolution as μ increases up to μ = 1/2.

Keywords

Restricted three-body problem Periodic solutions Two-parameter families 

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Copyright information

© Springer Science+Business Media B.V. 2006

Authors and Affiliations

  • Alexander D. Bruno
    • 1
  • Victor P. Varin
    • 1
  1. 1.Mathematical DepartmentKeldysh Institute of Applied MathematicsMoscowRussia

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